For each of the following vector fields, find its curl and determine if it is a gradient field. (a) F = 5yzi+ (5xz + z²) j + (5xy+2yz) k curl F = 0 Fis a gradient field (b) G = (10xy+x³) i+ (5x² + z²) }+ (2yz — 4z) k curl Ğ Ġ is a gradient field (c) H = 5yzi+ (22- 5xz) + (5xy + 2yz) k curl H = [ His not a gradient field

Suppose T: R¹→P3 is a linear transformation whose action on a basis for R4 is as follows: -3 a C = II 9x²-12x+6 T 0 0 Describe the action of T on a general vector, using x as the variable for the polynomial and a, b, c, and d as constants. Use the '^' character to indicate an exponent, e.g. ax^2-bx+c. -4x²+4x-4 T −4x³−8x²+14x−14 T 2x³-6x²+5x-2

Let F and G be vector fields with differentiable components. Express curl (F x G) in term of div and dot products.

Compute the curl of the vector field F = - (x +3y) 7 + (y – 3z) } – (z + 3x) k curl %3D

Can I get help with sketching this vector field

Please make a graph of the gradient vector field

Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector field. (5x + 7y, 8x + 5y, 0) = where the first vector in the sum is curl free and the second is divergence free. (For this problem, enter your vectors with angle-bracket notation: , not in i jk-notation.) +

For what values of b and c will F = (y2 + 2czx)i + y(bx + cz)j + ( y2 + cx2)k be a gradient field?

Find the curl of the vector field F = curl = k .

Discuss what it means to say that the curl of a vector field is independent of a coordinate system. Explain how we know this to be true